Real-time dynamics of the confining string

Transcription

1 Real-time dynamics of the confining string A Dissertation Presented by Frashër Loshaj to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Physics Stony Brook University August 2014

2 Stony Brook University The Graduate School Frashër Loshaj We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation. Dmitri Kharzeev Dissertation Advisor Professor, Department of Physics and Astronomy George Sterman Chairperson of Defense Distinguished Professor, Department of Physics and Astronomy Xu Du Assistant Professor, Department of Physics and Astronomy Larry McLerran Senior Scientist Brookhaven National Laboratory, Nuclear Theory Group This dissertation is accepted by the Graduate School. Charles Taber Dean of the Graduate School ii

3 Abstract of the Dissertation Real-time dynamics of the confining string by Frashër Loshaj Doctor of Philosophy in Physics Stony Brook University 2014 Quantum chromodynamics (QCD) describes the interaction of quarks and gluons, which are charged under the color group. Due to confinement of color charge, only colorless hadrons are observed in experiment. At very short distances (hard processes), perturbation theory is a valid tool for calculations and predictions can be made which agree well with experiment. Confinement, which is not yet understood from first principles, is important even for hard processes, because after the perturbative evolution is finished, the final colored particles combine to create the final state hadrons. There are many effective theories of confinement developed over the years. We will consider the Abelian projection; the gauge theory becomes Abelian-like and the theory contains magnetic monopoles. Confinement happens due to the dual Meissner effect, where dual in iii

4 this case means the roles of the electric and magnetic fields are reversed. The field between charges resembles that of an Abrikosov- Nielsen-Olesen vortex or string. Based on the Abelian nature of the confining string, because fermion zero modes are localized along the vortex and by considering very energetic jets, we assume that the dynamics along this string is described by massless quantum electrodynamics in 1+1 dimensions. This theory shares with QCD many important properties: confinement, chiral symmetry breaking, theta-vacuum, and is exactly soluble. We use the model to compute the fragmentation functions of jets in electron-positron annihilation and after fixing two adjustable parameters, we study the modification of fragmentation functions of jets in the QCD medium. We address an important puzzle in hadron scattering: the soft photon yield in processes with hadrons in the final state is much larger than what is expected from the Low theorem. We find that soft photons produced from currents induced during the real-time dynamics of jet fragmentation can contribute in the enhancement of photons. We compare the result with the recent DELPHI measurements and a reasonable agreement is found. Finally, assuming the QCD string to be thin, we address the observed phenomenon in recent lattice studies of partial chiral symmetry restoration along the string. Our result agrees well with the data. iv

5 Contents List of Figures Acknowledgements vii x 1 Introduction 1 2 Abelian model of the QCD string Abelian Projection Abrikosov-Nielsen-Olesen (ANO) Vortex The Schwinger model The anomaly and Dirac sea - spectral flow Bosonization The θ-vacuum String tension and chiral symmetry breaking Confinement and screening Discussion Fragmentation functions of jets in vacuum Space-time picture of jet evolution QCD string breaking and particle creation Fragmentation functions in e + e hadrons Discussion Jets in medium - Energy Loss Landau-Pomeranchuk-Migdal Effect (LPM) in perturbation theory v

6 4.2 In-medium jet fragmentation without gluon emission In-medium fragmentation with non-static sources and gluon emission Transverse-momentum difference between Au+Au and p+p Discussion Anomalous soft photon production Review of Low theorem Oscillations of electric and axial charge in quark fragmentation The soft photon production due to the axial anomaly Phenomenology of soft photon production Discussion Further applications Probabilistic fluctuation of the thin string Chiral symmetry restoration Anomaly inflow Electric charge distribution of the jet Discussion Conclusion 91 A Conventions and solution of equation of motion 93 A.1 Conventions A.2 Green s function A.3 Solution of inhomogeneous Bessel equation A.4 Compactness of the scalar field B U(1) magnetic monopoles 98 Bibliography 102 vi

7 List of Figures 1.1 Rectangular Wilson loop Space-time diagram of the motion of the massless relativistic string Breaking of the QCD string String breaking in two points Electric flux tube formed between a quark-antiquark pair moving back to back String breaking by formation of a massless pair of quark-antiquark. The electric field between the created pair does not vanish Numerical solutions of equations (2.29). Solid line is f, dashed line is a and e times the magnetic field B is the dotted line Level crossing in background electric field Field configuration after a pair production Ground state energy in units of length L Di-jet event in high energy scattering Gluon emission from a quark Quark and antiquark moving in opposite direction Scalar field φ as a function of the spatial coordinate z for m = 0.6 GeV and t = 10 fm String breaking by creation of pairs of charge-anticharge Electric field E(t 38/g, x) for self-consistent fermions (solid line) and external charges (dashed line) for g/m = 100, with E(t, x) = E(t, x) vii

8 3.7 The spectrum of charged hadrons in e + e annihilation at s = 201 GeV; solid line is obtained from (3.31) Rapidity distribution of charged pions in e + e annihilation experiment, with center of mass energy s = 29 GeV. Solid curve is calculated from (3.31) Incoherent scattering of the energetic quark with the medium constituents Coherent scattering of the quark with medium constituents (LPM regime) The color flow in the jet interactions inside the quark-gluon medium Ratio of in-medium to vacuum fragmentation functions. The length of the medium is fixed at 4 fm, the jet energy is E jet = 100 GeV. Solid line: the first scattering occurs at t 1 = 1 fm (assumed thermalization time), and subsequent scatterings occur with time spacing of t = 1/m = 0.3 fm. Dashed line: double scattering with t 1 = 2 fm and t 2 = 4 fm ( t = 2 fm). Dot-dashed line: four scatterings with t = 1 fm, t 1 = 1 fm. Data points are from the CMS Collaboration (see text). Open (filled) circles are for the leading (subleading) jet Energy loss as a function of jet energy. The lines correspond to the parameters in the caption of Fig In-medium scattering of the jet accompanied by an induced gluon radiation outside of the medium. Left: the Feynman diagram. Right: the corresponding color flow The ratio of in-medium and vacuum fragmentation functions for p jet = 120 GeV. The first scattering occurs at t 1 1 fm, which is the assumed thermalization time. The length of the medium is L = 5 fm. The curves correspond to mean free paths of λ = 0.57, 0.4 and 0.2 fm from top to bottom respectively. The data points are from the CMS Collaboration In-medium scattering of a gluon radiated from the original jet. 63 viii

9 4.9 D AA defined in (4.21) and (4.23) for jet energy of 20 < p jet < 40 GeV. Black dots and shaded areas show experimenal data, jet energy scale, v2/v3 and detector uncertainties respectively, taken from the STAR Collaboration paper (see text); solid line interpolates between calculated values of D AA from (4.23) Scattering of a charged (solid line) and a neutral (dashed line) particle Fermion and antifermion moving back-to-back Electrically charged current, resembling a quantum wire, coupled to a dimensional electromagnetic field The soft photon yield as a function of the jet momentum. The circles are the DELPHI Collaboration data (see text), and the squares represent the calculations of soft photon production based on the Low theorem (see (5.1)). The solid line is our result Thin string fluctuating in the transverse plane The longitudinal chromoelectric field between two static charges as given by (6.9) The chiral condensate around a confining flux tube as a function of the transverse distance. The solid line is the result of our model based on (6.8). The squares are the lattice data (see text) Feynman diagram contributing to the vacuum current Inflow current in the di-jet event B.1 The choice of the gauge potential in different patches of space. 100 ix

10 Acknowledgements I would like to thank my advisor Dmitri Kharzeev for his guidance and support during my last years as a PhD student and for his help in finishing the projects on which this thesis is largely based on. I thank the members of the committee Xu Du, Larry McLerran and George Sterman for agreeing to participate in the defense. Over the past few years I have benefited from numerous discussions with professors of the Nuclear Theory group Edward Shuryak, Derek Teaney and Ismail Zahed. I would like to thank also Tigran Kalaydzhyan, Wolfger Peelaers and Ho-Ung Yee for many useful discussions. I am indebted to Dmitri Kharzeev and George Sterman for reading earlier versions of the thesis and for many useful comments. I am grateful to Laszlo Mihaly for helping me transfer from the electrical engineering department and for giving me the opportunity to pursue a PhD degree in physics. x

11 Chapter 1 Introduction At very short distances (high energies), due to asymptotic freedom [1, 2], QCD can be viewed as a theory of weakly coupled quarks and gluons. On the other hand, at distances of the order of the proton size, namely 1 fm, the theory becomes strongly coupled and degrees of freedom are hadrons instead of quarks and gluons. The latter do not exist as asymptotic states due to the formation of chromoelectric flux tube or string between color sources - confinement. QCD is not solved yet at these low energies and confinement is not fully understood from first principles. There are, however, indirect indications for its existence. For example, an order parameter that is useful in describing the phases of QCD is the Wilson loop operator [3], defined as { [ ]} W (C) = Tr P exp i A µ (x)dx µ. (1.1) C P stands for path ordering, C is a closed contour of integration and A µ is the gauge field. We can imagine the contour as the trajectory of a quark in space-time. For example, in the case of a very heavy quark-antiquark pair, separated by distance R, existing for time T, the contour will be a rectangle as shown in Fig For a very large value of time T, the vacuum expectation value of the Wilson loop operator becomes W (C) exp[ V (R)T ], (1.2) 1

12 R Figure 1.1: Rectangular Wilson loop. where V (R) is the static potential between the quarks. If the potential is linearly confining, namely V (R) = σr, where σ is the string tension, the Wilson loop operator takes the value W (C) e σrt. (1.3) This is known as the area law and is an indicator of confinement. The existence of light quarks in the theory, causes the potential between quarks to be screened, in other words, the string stretched between them breaks and the Wilson loop operator obeys the perimeter law instead. These properties can be checked for example in lattice QCD. Another feature of QCD is its complicated and rich vacuum structure. The masses of the light quarks in QCD are very small compared to the characteristic mass scale M 1 GeV (the scale below which the perturbative QCD is invalid). Since the interaction between quarks is mediated via a vector exchange, in the limit of zero mass for the quarks, the theory is chirally symmetric - quarks with different chirality do not transform between each other. This symmetry is absent in the spectrum of hadrons, therefore it must be spontaneously broken. From the Goldstone theorem [4], the spontaneous symmetry breaking is accompanied with the appearance of massless bosons (Goldstone bosons) in the spectrum. If m u, m d 0 (SU(2) symmetry), then we have the breaking SU(2) L SU(2) R SU(2). In this case, there should be three Goldsone bosons, which can be identified with the triplet of the π mesons. The spontaneous breaking of the chiral symmetry is accompanied with nonzero value of certain condensates. For example, from the Gell-Mann-Oakes-Renner 2

13 relation [5] 0 qq 0 = 1 m 2 πfπ 2, (1.4) 2 m u + m d where m π and f π are the mass and the decay constant of the pion respectively. Using the numerical values of these parameters (see for example [6]), we get 0 qq 0 (257 MeV) 3. (1.5) The interpretation of this result is that a nearly massless quark propagating in the QCD vacuum dynamically obtains a finite mass. Another property is, for example, the large scale fluctuations of the gauge field. This can be illustrated by the so-called gluon condenstate, given by [6] 1 32π 2 0 F µνf µν 0 (200 MeV) 4. (1.6) In quenched QCD, the limit of infinitely heavy quarks, one can use the string tension as an order parameter of confinement-deconfinement transition. On the other hand, in the chiral limit (zero fermion mass), the chiral condensate can be an appropriate order parameter of the chiral phase transition. It is well known however that both string tension and chiral condensate are present in both phases, which suggests that we are dealing with a crossover instead of a phase transition. Some nonperturbative aspects can be addressed analytically, as is the case in the so called Shifman-Vainshtein-Zakharov (SVZ) sum rules [7], which is a method for extracting the long distance current correlation functions using the values of certain condensates. For the most part of this thesis we will be interested in nonperturbative aspects of high energy processes. We should mention that perturbative QCD (pqcd) calculations, over the years, have shown remarkable agreement with experimental results. This success can be attributed to properties like factorization (see [8]) - where one can treat independently the hard part of high energy hadron scattering from the soft part, and local parton hadron duality (LPHD) [9] - where distributions of final quarks and gluons (partons) in QCD cascades, seem very similar to the final hadron distribution in experiment. 3

14 Let us consider a simplified picture of a high energy event. The hard part of the process can be treated perturbatively, where the dynamical degrees of freedom are quarks and gluons. As we go to large distances, or small momenta, the transition between quark and gluon to hadron degrees of freedom takes place. This process is known as hadronization. There are many models that are used to describe the transition of quarks and gluons to hadrons. We will review very briefly below the so-called Lund model, which is based on the semiclassical treatment of the QCD string. A pedagogical review of the subject is given in [10]. We will mostly follow [10 12]. The starting point of the string model is the fact that the potential between q q grows linearly with distance, V (r) = κr, where from experiment (for example hadron mass spectroscopy) it is known that κ 1 GeV/fm 0.2 GeV 2 (at small distance between quarks, a Coulomb potential is also present, but it is neglected, because it does not affect the fragmentation, only the properties of the final hadrons). In order for the description to be Lorentz covariant and causal, the most straightforward way is to require that the endpoints of the strings be massless and to neglect transverse degrees of freedom. The string formed this way is referred to as the massless relativistic string. Consider for example a quark q and antiquark q moving back to back, with a linear potential stretching between them. As the quarks move, the energy of the string increases and when it equals the initial kinetic energy of the system, the quark-antiquark abruptly change direction and start moving back (see Fig. 1.2). The mass of the system is proportional to the area spanned by the endpoints of the string. After including dynamical fermions, the string should break, splitting into two color singlet states. This is illustrated in Fig The thick horizontal lines indicate the electric field stretched between the pairs. The two systems move apart and the electric field between the two vanishes as indicated in Fig If the invariant masses of the new strings are large, they continue to break, until the masses reach values comparable to masses of hadrons. Thus, in the process of pair creation, the original string breaks into n strings, which form the primary hadrons (in Fig. 1.3 we illustrate a formation of a hadron in the 4

15 t Figure 1.2: Space-time diagram of the motion of the massless relativistic string. Figure 1.3: Breaking of the QCD string. 5

16 far left). The time ordering of the breaking is irrelevant because the points where the string breaks are space-like separated. This can be seen as follows. Since the string has a constant tension κ, de dz = dp z dt = κ. (1.7) Now assume that the string breaks at points 1 and 2, as shown in Fig. 1.4, with coordinates (t 1, z 1 ) and (t 2, z 2 ) respectively. The energy and momentum 1 2 Figure 1.4: String breaking in two points. of the system between points 1 and 2 are given by E = κ z = κ(z 1 z 2 ), p z = κ t = κ(t 1 t 2 ). (1.8) The total mass is E 2 p 2 z = m 2 = (z 1 z 2 ) 2 (t 1 t 2 ) 2, (1.9) and is positive only for space-like separation of the points. The breaking then occurs independently. This enables one to define an iterative procedure, which guaranties that the produced hadrons are on mass shell. It should be pointed out that heavy quarks like charm and bottom are not produced as the string breaks, but they can only be at the endpoints of the initial string. This can be argued by using the Heisenberg and Euler theory of pair creation in a strong field [13] (for a review see [14]). The probability for creating a pair of particles with mass m and transverse momentum p is 6

17 given by ( ) exp πm2 = exp κ ( πm2 κ ) ( ) exp πp2, (1.10) κ where m = p 2 + m2. We see that this expression implies suppression of production of heavy pairs, namely u : d : s : c 1 : 1 : 0.3 : The Lund model has proven to be very successful in describing many experimental results. Its usage however, is quite limited when the quantum dynamics of the string breaking is required. The quark gluon plasma formed in heavy ion collisions is strongly coupled and its effect on jets involves nonperturbative physics. One needs a dynamical theory of confinement in this case (see chapter 4). Another example would be in the case of photon production from the real-time dynamics of jet fragmentation (see chpater 5). Despite all the successes of perturbative QCD, there is a need to understand better the nonperturbative aspects of it. A lot of data from the soft-physics part from LHC, for example, may challenge the standard fragmentation picture. It should also be questioned if this picture applies in heavy ion physics as well. Another reason to understand the fragmentation of quarks and gluons comes when studying ultra high energy cosmic rays. The extrapolation of the known models from collider energies to these processes seems to be challenged (see for example [15]). In this thesis, using a very well known exactly soluble theory, which shares with QCD many key properties, we will develop a simple model of dynamical string breaking. More details will be given in chapter 2. We will give the basic idea here, but first we have to say something about another model of confinement, the dual superconductor theory. The mechanism of confinement in terms of the dual Meissner effect is due to Nambu, Mandelstam, t Hooft and many others (see for example [16 18]). The basic idea is as follows. Since quarks contain chromoelectric charge, if a quark q and an antiquark q are put at a distance R from each other, there will be a chromoelectric field stretching between them. If these particles are embedded in a non-superconducting medium, then a Coulomb like field is stretched in between. On the other hand, if the medium is a superconductor, then the field between quark and antiquark will resemble a flux tube, which is the result of 7

18 Meissner effect (for type II superconductors). The field cannot be expelled completely, because the flux should be conserved. As the distance between particles increases, the flux tube becomes longer, but it maintains the same minimal thickness and the transverse profile is unchanged. These flux tubes are similar to the Abrikosov-Nielsen-Olesen vortices, which we will review in the next chapter. The effect we described so far is referred to as the dual superconductivity, because as we can see the roles of the electric and magnetic fields are interchanged, compared to the usual superconductor. For the usual type II superconductor, the electric charges condense, into the so called Cooper pairs, and magnetic flux tubes are formed between magnetic monopoles. In our case, the flux tubes are electric and in turn magnetic monopoles should condense. Furthermore, it is assumed that the field between the quark and antiquark is Abelian. In this case the dual superconductor is described via the Landau-Ginzburg model. This Abelianization can be achieved, for example, through the method of Abelian projection, which we will also briefly review in the next chapter. Based on the dual superconductor theory of confinement, we assume that the dynamics along the QCD string is Abelian. The electric field between charged particles is similar to that of the vortex in the Landau-Ginzburg theory. Moreover, as we will see in the next chapter, if one introduces fermions to this theory, the zero modes can be seen to be localized along the core of the vortex. For the most part of this work, we consider very energetic jets, therefore the longitudinal scale, along the jet axis, is much larger than transverse. We then assume a dimensional dynamics. Our assumption therefore will be that the dynamics of fermions on the string is described by massless QED in 1+1 dimensions, also known as the Schwinger model. The theory is exactly soluble. String breaking due to massless fermions is similar to the Lund model as shown above. We will give many details in subsequent chapters, but let us mention an important difference between the semiclassical treatment of the Lund model and our case. Let us consider again a quark-antiquark pair moving back to back as in Fig In the Lund model model, as we saw above, the string breaks due to the creation of massless pairs and the electric field between them 8

19 Figure 1.5: Electric flux tube formed between a quark-antiquark pair moving back to back. vanishes. In the Schwinger model the situation is different. As the new pair is created, the field between them is not completely canceled (see Fig. 1.6). Instead, as new pairs are created, they create a dynamical fluctuation of the field. In dimensions, the chirality of fermions is determined by their Figure 1.6: String breaking by formation of a massless pair of quark-antiquark. The electric field between the created pair does not vanish. direction of motion and charge. The right moving fermion and a left moving antifermion have the same chirality. This means that the system in Fig. 1.5 has two units of chirality, which is encoded in the electric field due to the index theorem in dimensions N R N R N L = g 2π d 2 x ɛ µν F µν. (1.11) and N L are the number of right and left handed particles respectively (in our conventions, the system has two units of right chirality). The axial anomaly is precisely given by the integrand of (1.11), therefore it determines the particle creation. We will discuss more about the role of the anomaly in particle creation in Chapter 5. The following is the outline of this thesis. In chapter 2 we review the so-called Abelian projection of gauge theories. The diagonal part of the gauge field remains massless and the theory can be thought of as having a U(1) N 1 gauge invariance. In the simplest case of SU(2) it is shown that singular points of the transformed field can be interpreted as containing magnetic monopoles. From these considerations, we assume that 9

20 the QCD string resembles a dual Abrokosov-Nelsen-Olesen vortex. If one introduces fermions into the theory, it is well known that zero modes will be confined along the core of the string. We therefore conjecture that the dynamics of massless fermions along the QCD string is described by QED in dimensions, also known as the Schwinger model. The theory is exactly soluble. We illustrate this in a simple way using bosonization. We review this theory and notice that it shares with QCD many important features, including confinement, screening of charge, chiral symmetry breaking, anomalies and θ- vacuum. In chapter 3 we use the Schwinger model to study the real-time dynamics of jet fragmentation. Our aim is to calculate analytically the fragmentation function of the quark. The coupling of the theory has dimension of mass. In the bosonized version of the theory the coupling is interpreted as the mass of the produced hadron in the fragmentation of QCD string. We introduce another parameter Q 0, whose physical interpretation is that of the scale at which the perturbative cascade stops. We fit these parameters by comparing with data from e + e annihilation to hadrons. Our result, by choosing the hadron mass to correspond the σ meson mass and Q 0 2 GeV, agrees reasonably well with the data. The most significant discrepancy is seen for soft hadrons and this is due to the fact that we neglect the produced gluons in the perturbative cascade. In chapter 4, we adopt the model to include the effect of QCD medium, produced in heavy-ion collisions, in jets. We consider a very simple case of a static medium. As the jet propagates, it scatters off the static sources, which are placed at equal distance from each other. After each scattering, the quark rotates in color space, creating many independent antennae giving off radiation, by which the jet loses energy. This scenario seems to explain well the experimental observation of soft hadron enhancement for in-medium jet fragmentation. We then consider the case where the jet exchanges momentum with the medium and gives off a final gluon outside the medium. As a result of partial screening of the color charge of the jet from the final gluon, we obtain the observed suppression for intermediate momentum hadrons. A long lasting puzzle in high energy hadron scattering is the phenomenon 10

21 of anomalous soft photon production. It corresponds to the observed fact that in many experiments the soft photon yield cannot be explained from the Low theorem - the observed yield is up to five times higher than expected. The phenomenon is only observed in processes with hadronic final states. It is reasonable to assume that part of the enhancement of photons can come from the real-time dynamics of QCD string breaking in the process of jet fragmentation. We use the model introduced in the previous chapters. By assuming a non-zero width of the hadron propagator and taking into account the fluctuations of the string, we can describe the recent DELPHI measurements in photon enhancement. This will be the focus of chapter 5. Recently, a lattice measurement of the chiral condensate was performed in the presence of static quarks (QCD string stretched between them). It was shown that the chiral condensate decreases at the core of the string, which signals partial chiral symmetry restoration. In chapter 6 we assume a thin string between the quarks, with a probabilistic fluctuation in the transverse plane. Assuming the same energy density per unit length in the full dimensional string and the dimensional case, we derive the probability distribution. In order to compare with the data, we have to fix the effective width of the string, which is shown to be of the order of the lattice spacing. We give a summary of all the results of the thesis and mention some of the future prospects in chapter 7. 11

22 Chapter 2 Abelian model of the QCD string In this chapter we will argue that the dynamics along the string formed between color charges, if we consider light fermions in the theory, is described by a dimensional, Abelian model. In order to do that, we first review some facts about the Abelian projection of SU(N) gauge theory (we only consider SU(2) for simplicity). The Abelian projected theory contains magnetic monopoles. These monopoles can condense, giving rise to a dual superconductor, described by the Landau-Ginzburg theory. Duality here means that the roles of the magnetic and electric fields are interchanged. Between colored sources, a (chromo-) electric field flux tube is formed and confinement ensues. This scenario is similar to the case of the usual superconductor where the magnetic monopoles are confined. In later chapters we use this model to study the real-time dynamics of jet fragmentation. The jets under consideration are very energetic; the longitudinal (along jet axis) scale is much larger than transverse, therefore we neglect the transverse degrees of freedom in our treatment. We propose that the dynamics of jet fragmentation at late stages, where the perturbative QCD emission ends, is described effectively by massless QED 2, also known as the Schwinger model. We review below some basic facts about this model and show that it shares with QCD many important aspects, needed to describe jet fragmentation. These include confinement, chiral 12

23 symmetry breaking, anomalies and θ-vacuum. 2.1 Abelian Projection In this section we review the so-called Abelian projection (also known as Abelian gauge fixing). The basic ideas shown here have been introduced in the seminal papers by Polyakov [19] and t Hooft [20, 21]. For a review see [22, 23]. The aim of the Abelian projection is to find a gauge where the SU(N) gauge field A a µt a is diagonal (T a are the generators of SU(N)). This would reduce the non-abelian theory to an Abelian one. This however cannot be achieved - not all components of the gauge field can be diagonalized simultaneously. We can instead proceed in the following way. Let s consider a scalar in the adjoint representation of the gauge group φ(x) = φ a (x)t a. (2.1) The scalar can be constructed out of the fields of the gauge theory, for example φ = G 12, where G µν is the field strength. The goal of this procedure is to choose a gauge where the field φ(x) is diagonal, in other words Ω(x)φ(x)Ω (x) = diag(λ 1 (x), λ 2 (x),..., λ N (x)). (2.2) This gauge is called the Abelian gauge and in general it depends on the choice of φ(x). We will discuss below how the monopoles appear in this gauge for the case of SU(2). The gauge fixing (2.2) now becomes ( ) φ(x) Ω(x)φ(x)Ω λ 0 (x) =, λ = (φ 1 ) 2 + (φ 2 ) 2 + (φ 3 ) 2. (2.3) 0 λ We notice that the eigenvalues are degenerate when λ = 0, which means that all components of φ(x) vanish. Let s consider the points where this happens, namely φ 1 (r 0 ) = φ 2 (r 0 ) = φ 3 (r 0 ) = 0 (2.4) 13

24 We have three equations and three unknowns, which are the components of the vector r 0. At the point r 0, it is not possible to define the gauge, and we will see that the gluon field develops a singularity at that point. Let us Taylor expand the field φ(r), where r = (x 1, x 2, x 3 ), at the vicinity of the point r 0 φ(r) = φ a (r)t a = T a C ab (x a x 0b ) (2.5) where C ab = φa (r) x b xa=x 0a, (2.6) and x 0a are the components of r 0. We see that in this new coordinate system, defined by the transformation C ab, the field takes the form φ(r) = x a T a, x a = C ab (x b x 0b ). (2.7) The scalar has now the form of the hedgehog field. From now on, we assume to be in the new frame and we drop the primes. Let (r, θ, φ) be the spherical coordinates of the vector r. The hedgehog field φ(x) = x a T a takes the form φ(r) = x a T a = T 1 r sin θ cos φ + T 2 r sin θ sin φ + T 3 r cos θ (2.8) In the case of SU(2), the generators are given by T a = 1 2 σa, where σ a are the Pauli matrices (see (A.4)). We then have ( ) φ(r) = r cos θ e iφ sin θ. (2.9) 2 e iφ sin θ cos θ It is easy to see that the gauge transformation ( e iφ cos θ sin θ 2 2 Ω(θ, φ) = sin θ e iφ cos θ 2 2 ), (2.10) diagonalizes the field φ(x). In other words, ( Ω(x)φ(x)Ω (x) = r ) = rt 3. (2.11) 14

25 We next check how the gluon field looks like in this gauge ( A µ A a µt a Ω A µ + 1 ) ie µ Ω, (2.12) where e is the coupling constant. To check how the spatial components of the gauge field transform, we first have to compute Ω Ω = ˆr (Ω r ) ( Ω + ˆθ Ω 1 r ) ( θ Ω + ˆφ Ω 1 r sin θ ) φ Ω. (2.13) From the explicit expression of the transformation matrix (2.10), we get Ω r Ω = 0, (2.14) Finally, Ω θ Ω = ie iφ T 2, (2.15) Ω φ Ω = i(1 + cos θ)t 3 + i sin θ cos φt 1 i sin θ sin φt 2. (2.16) 1 ie Ω Ω = 1 ( ˆθ 1 e r eiφ T cos θ ˆφ r sin θ T 3 + ˆφ 1 ) r (cos φt 1 sin φt 2 ). (2.17) We can write the result in the following form A = A a T a = A a RT a 1 e ˆφ 1 + cos θ r sin θ T 3, (2.18) where A R is the regular part of the gauge field, whereas the second term is singular as θ 0 and has the form of the potential of a magnetic monopole (see Appendix B). Comparing this result to the expression for the potential of a magnetic monopole, we get the magnetic charge g = 1 e. (2.19) 15

26 The Dirac quantization condition 2eg = n, where n is an integer, is fulfilled. We have thus shown that at the points where the Abelian gauge is ill defined (or singular), the diagonal (Abelian) part of the gauge field behaves as if at those points there are magnetic monopoles with magnetic charge g = 1. It e can be shown that the theory contains an Abelian gauge field and massive scalars. In the case of SU(N), this procedure leads to a theory with U(1) N 1 symmetry and charged massive scalars that contain two types of charges (from two different U(1) sectors) [22, 23]. Magnetic monopoles also appear at points where the gauge field is singular. It is shown that indeed in this gauge, magnetic monopoles exist and if they condense, confinement can be viewed as appearing due to the dual Meissner effect. 2.2 Abrikosov-Nielsen-Olesen (ANO) Vortex Let us now see how the vortex is formed by considering an Abelian-Higgs model. The vacuum expectation value of the Higgs boson plays the role of the order parameter, which in the case of the ordinary superconductor is the condensate of Cooper pairs and in the dual superconductor picture is the condensate of magentic monopoles. It is well known that this theory contains vortex solutions, which were first found by Abrikosov [24] for the case of superconductivity and later were generalized to the relativistic case by Nielsen and Olesen [25]. We consider the Lagrangian in dimensions L = 1 4 F µνf µν (D µφ) (D µ φ) λ 4 ) 2 ( φ 2 µ2, (2.20) λ where φ is a complex scalar field, D µ = µ φ+iea µ φ and e is the U(1) coupling constant. It is clear that the scalar field acquires a vacuum expectation value φ = µ 2 λ = v, (2.21) and the U(1) symmetry is spontaneously broken. The Higgs field φ(x) and the gauge field acquire masses m φ = 2λv and m A = ev respectively. The 16

27 Lagrangian is invariant under the gauge transformations φ(x) e ieλ(x) φ(x) A µ (x) A µ (x) µ Λ(x). (2.22) We can define, for any smooth configuration of φ, the winding number N[C] = 1 dx (arg φ), (2.23) 2π C where C is the circle at x =. If φ is nonzero everywhere on the circle C, then this quantity is well defined and measures the total rotation of the phase of φ. It follows from the definition (2.23) that this quantity is an integer. It is easy to see that for Λ(x) single-valued everywhere, the winding number is gauge invariant. If we take the gauge field at large distances to be pure gauge (this means that the field strength vanishes) A(x) = 1 α(x), (2.24) e the winding number is given by N[C] = n = 1 dx α = e dx A = e 2π C 2π C 2π d 2 xb = e Φ, (2.25) 2π where B = F 12 is the magnetic field in the z direction and we have used Stokes theorem to get the fourth equality. The last surface integral, is over the area bounded by the curve C. This tells us that the flux is quantized Φ = 2πn e. (2.26) We will look for static solutions and set A 0 = 0. Assuming the most general rotationally symmetric solution and requiring that the solution is invariant under reflection about the x-axis (the scalar field gets complex conjugated), 17

28 we adopt the following ansatz φ(x) = ve iθ f(evr) A j (x) = ɛ jkˆx k a(evr), (2.27) er with functions f and a real. This solution has winding n = 1. It can be checked that the solution (2.27) has finite energy and it scales like ( ) λ E = v 2 F e 2 ( ) = m2 φ λ 2λ F, (2.28) e 2 where the function F is of the order of unity. The equations of motion, in terms of f and a are f + 1 u f a 1 u a + (1 a)f 2 = 0, (1 a)2 u 2 f + λ e 2 (1 f 2 )f = 0, (2.29) where u = evr and the primes denote derivatives with respect to u. requirement that the solution is regular at the origin gives the condition The f(0) = a(0) = 0, (2.30) whereas requiring finite energy solution gives us f( ) = a( ) = 1. (2.31) Having set the boundary conditions, the equations can be solved numerically. We show the solutions for λ/e 2 = 1/2 in Fig. 2.1, which was taken from [26]. We see that the magnetic field is localized around the core and falls exponentially away from it. In the dual picture, the magnetic field is replaced by the electric field and between electric charges an electric flux tube is formed. Let us mention some facts about the multi vortex solution, or n > 1. In this case, we cannot argue that the minimal energy configuration necessarily gives a 18

29 Figure 2.1: Numerical solutions of equations (2.29). Solid line is f, dashed line is a and e times the magnetic field B is the dotted line. static solution. In other words, this means that not all zeros of φ are located at the origin. Qualitatively, in order to see if the n vortex solution dissociates into n unit vortices, we should consider the interaction between two well separated vortices. There are two types of interactions, the electromagnetic and the scalar. The sign of the force depends on which of these interactions dominates. The electromagnetic force is repulsive, whereas the scalar force is attractive. Both forces fall off exponentially at large distance, therefore the dominating force is the one falling off more slowly. We have an attractive force for m φ < m A, and repulsive if m φ > m A [26]. In the superconductor phenomenology, the first case corresponds to Type I superconductor, whereas the second to Type II. We now add fermions to a dimensional Abelian-Higgs model and see what are the consequences. We assume that the vortex is situated along the z-axis. The fermions are given by four component spinors ψ, satisfying γ 5 ψ R = ψ R, (2.32) γ 5 ψ L = ψ L, (2.33) where γ 5 = iγ 0 γ 1 γ 2 γ 3 and ψ L,R = 1 2 (1 γ5 )ψ. We introduce the fermions to 19

30 the theory by adding L f = i ψ R γ µ D µ ψ R + i ψ R γ µ D µ ψ R gφ ψ L ψ R gφ ψr ψ L, (2.34) to Lagrangian (2.20). g is real and is the fermion-scalar coupling constant and D µ ψ = ( µ + iqa µ ) ψ. The gauge symmetry requires that e = 2q, where e is the charge of φ. The Lagrangian has the symmetry ψ e iβγ5 ψ, φ e 2iβ φ, (2.35) and doesn t have a bare (independent of φ) mass term. The Dirac equation can be written as iγ µ D µ ψ L gφψ R = 0, iγ µ D µ ψ R gφ ψ L = 0. (2.36) We look for solutions ψ (0) L,R, which depend only on the transverse radial direction r = (x 1 ) 2 + (x 2 ) 2. We also assume that iγ 1 γ 2 ψ (0) L = ψ (0) L, iγ 1 γ 2 ψ (0) R = ψ (0) R (2.37) Let us consider the first of equations (2.36). We can write it as i ( γ 1 D 1 + γ 2 D 2 ) ψ (0) L gφψ(0) R = 0. (2.38) We multiply both sides by γ 1 and recall that (γ 1 ) 2 = 1. We get ( ) D1 γ 1 γ 2 (0) D 2 ψ L igφγ1 ψ (0) R = 0. (2.39) We use the property (2.37), relations (2.27) and [ D 1 ± id 2 = e ±iθ r ± i r θ ± a(r) ], (2.40) 2r 20

31 to get [ r + a(r) ] ψ (0) L 2r igvf(r)γ1 ψ (0) R = 0. (2.41) Similarly, for the second of equations (2.36), we have [ r + a(r) ] ψ (0) R 2r igvf(r)γ1 ψ (0) L = 0. (2.42) Note that the sign before a(r) is the same in both equations, despite the fact that there is a sign difference in relations (2.27). This is due to the fact that ψ L and ψ R have opposite charges. The solution of equations (2.41) and (2.42) is given by and r [ ]} ψ (0) L { = C exp dr gvf(r ) + a(r ) χ (2.43) 0 2r χ is a constant, left-handed spinor, which satisfies ψ (0) R = iγ1 ψ (0) L. (2.44) iγ 1 γ 2 χ = χ. (2.45) We assume that the general solution has the form ψ L = b(t, z)ψ (0) L (r), ψ R = b(t, z)ψ (0) R (r), (2.46) where b(t, z) is an arbitrary function of coordinates t and z. The Dirac equation above gives (γ γ 3 3 )b(t, z)χ = 0. (2.47) From the definition of γ 5 in dimensions, and multiplying the equation above by γ 0 γ 3, we notice that the spinor χ satisfies γ 0 χ = γ 3 χ. (2.48) The solution of equation (2.47) is given by an arbitrary function of t z. The 21

32 modes with definite energy have b = e iω(t z). The case ω > 0 corresponds to a right-moving particle with energy E = ω that moves with the speed of light along the vortex core (in the z-direction), whereas ω < 0 corresponds to an antiparticle with energy E = ω. Since the particles move with the speed of light, they must be massless. As we move away from the core, into the transverse plane, the fermionic field falls off exponentially. We see that along the vortex core we have an effective theory in dimensions, where massless fermions are coupled to a gauge field. In the case of an Abelian gauge theory, as the one considered here, the effective theory will be QED in dimensions (QED 2 ). Below, we review some facts about this theory. 2.3 The Schwinger model We saw above that confinement can be understood in terms of the dual Meissner effect. This means that due to condensation of magnetic monopoles, the vacuum can be considered as a superconductor, where compared to the usual superconductor, the roles of the electric and magnetic fields are reversed. Confinement happens as a result of monopole currents which expel the electric field from the superconductor, and due to the conservation of flux, the field between electric charges is stretched along a thin tube. In the following, we assume that the dynamics of fermions along these tubes is described in terms of massless QED 2. QED 2 (the Schwinger model) has been used a long time ago [27 29] as an effective theory capable to model the transformation of partons into hadrons at high energies [30] (see also [31, 32]). The high energy justifies the dimensional reduction to (1 + 1) dimensions even further, and the Schwinger model with massless fermions shares many key properties with QCD, including: i) the Higgs phenomenon (local electric charge conservation is spontaneously broken); ii) the spontaneous breaking of global chiral symmetry; iii) the screening of color charge (similar to the scenario of confinement for QCD with light quarks proposed by Gribov [33], see [34] for a review); iv) axial anomaly and the θ-vacuum. The main advantage of the Schwinger model with massless fermions (quarks) is that it is exactly soluble and can be used to investigate 22

33 e.g. the role of LPM effect at strong coupling. Of course, for our purposes we have to generalize it to allow for the rotation of quarks in color space (see Chapter 4) as they traverse the quark-gluon plasma (recently the effects of the color flow were addressed in [35]). The dynamics of QED 2 is described by the action S = d 2 xl (2.49) where the Lagrangian is given by L = 1 4 F µνf µν + ψi /Dψ (2.50) where, /D γ µ D µ, F µν = µ A ν ν A µ and D µ = µ + iga µ 1. This Lagrangian is invariant under gauge transformations ψ e iα(x) ψ, A µ A µ 1 g µα(x). (2.51) The fields have the following mass dimensions [A] = 0, [ψ] = 1 2, (2.52) which means that the coupling constant [g] = 1. (2.53) There are two classically conserved currents, the vector current j µ (x) = ψ(x)γ µ ψ(x), (2.54) and the axial current j µ 5 = ψ(x)γ µ γ 5 ψ(x). (2.55) 1 The index µ = 0, 1; A µ is the gauge field and g is the coupling constant. Other conventions are given in Appendix A 23

34 Their conserved charges are Q = dxj 0 (x), Q 5 = dxj 0 5(x). (2.56) After including the quantum corrections, if we want to preserve gauge invariance, then the axial current becomes anomalous. We will derive the axial anomaly in an intuitive way in the next section The anomaly and Dirac sea - spectral flow In dimensions it is very easy to see that the anomaly can be understood in terms of the Dirac sea (see for example [36 39]). Let s assume a compact spatial extension, namely x [ L/2, L/2]. We impose the following boundary conditions ( A µ t, x = L ) 2 ( ψ t, x = L ) 2 ( = A µ t, x = L ) 2 ( = ψ t, x = L ). (2.57) 2 By choosing an appropriate gauge, it is always possible to to remove x dependence for A 1, so A 1 = A 1 (t). We will assume that A 1 (t) is an external field, which changes adiabatically with time. We also assume that the Coulomb potential A 0 = 0, which is a reasonable approximation in the limit gl << 1 [38]. We still have some gauge freedom left. It is easy to see, by looking at the gauge transformation of fermions, that we are still left with A 1 A 1 + 2πn gl, (2.58) where n = 0, ±1,.... This means that the configuration space for the gauge field is a circle with length 2π. From the Lagrangian (2.50), we get the equation gl of motion (i/ g /A)ψ = 0. (2.59) 24

35 5Π L 3Π L Π L Π L ga 1 3Π L Figure 2.2: Level crossing in background electric field. L Using the conventions in Appendix A, we can write [ i ( t + σ 3 i )] x ga 1 ψ = 0. (2.60) The general solution to the equation (2.60), taking into account boundary conditions (2.57), can be written as (we omit an overall constant) ψ(t, x) 1 ( u(k) exp( ie k t) exp i 2π L L k ( k + 1 )) x. (2.61) 2 where E k is the energy of the kth mode. By plugging in the solution (2.61) to (2.60), we get the following energy spectra for left and right handed fermions ( ) E R k E L k = 2π L = 2π L k ga 1 2 ( k + 1 ) ga 1 (2.62) 2 We illustrate this in Fig 2.2. In this figure, the dashed and solid lines show the 25

36 levels for left and right components respectively. In the usual way, we fill up all negative energy levels and leave empty all positive ones, so at A 1 = 0 we have the vacuum state. As we change ga 1 = 0 to ga 1 = 2π (see for example L the case when k = 0), the levels L-levels move up, whereas the R-levels move down, in such a way that we arrive at the same ground state. This illustrates the above mentioned fact that the gauge field is compact. In process though, since the levels have shifted, we have created an L-particle (white circle) and an R-hole (black circle). The total electric charge is conserved, because the particle and the hole have opposite charge. On the other hand, the axial charge is just the difference between left and right handed fermions, namely Q 5 = N L N R. (2.63) In our conventions, a left-handed fermion has axial charge +1, whereas a righthanded antifermion has also axial charge +1. The total change in the axial charge is then Q 5 = 2. (2.64) If we write Q 5 = L π g A 1 (2.65) and divide both sides by t, we get Q 5 = L π g A 1. (2.66) Using (2.56) This gives L 0 dxj5 0 = g π 0 0 L 0 dxa 1. (2.67) 0 j 0 5 = g π 0A 1. (2.68) If we write the last equation in a Lorentz covariant way, we get µ j µ 5 = g 2π ɛµν F µν, (2.69) 26

37 which is the well known anomaly in dimensions. The anomaly can also be derived by considering the behavior at some ultra violet (UV) cutoff, as is the case in the perturbative treatment. When working with the Dirac sea, since the total energy and charge of the vacuum (we have infinitely many filled states) are ill-defined, one has to regularize the currents. The most often used procedure is the point splitting method of Schwinger, which preserves the gauge invariance. This method leads to the same result for the anomaly (see references given in the beginning of the section). To summarize, we have considered the Dirac sea and we have switched on, adiabatically, an external gauge field. As a result, the energy levels in the infinite sea shift; L-levels move up in energy and R-levels move down - spectral flow. This difference in energy levels gives the index of the Dirac operator. The total electric charge is conserved, but the axial charge changes by ±2 (depending on the sign of the gauge potential). By considering the UV behavior of the theory, we arrive at the same result. We haven t used any perturbative calculation and we see that that the anomaly is a purely topological effect Bosonization In one spatial dimension, the fermionic degrees of freedom can be expressed in terms of bosonic ones in an exact way. This correspondence is usually expressed as a duality between fermionic bilinears and a real scalar field. This procedure is known as bosonization. In the U(1) case the correspondence is given by the following relations [40, 41] ψ(x)iγ µ µ ψ(x) 1 2 µφ(x) µ φ(x) ψ(x)γ µ ψ(x) 1 π ɛ µν ν φ(x) ψ(x)ψ(x) c g π cos(2 πφ(x)) (2.70) where c = eγ and γ is the Euler constant. This is known as the Abelian 2π bosonization. Note that the scalar field has mass dimension [φ] = 0. We 27